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arXiv · 2411.10419

Lower bounds on the top Lyapunov exponent for linear PDEs driven by the 2D stochastic Navier-Stokes equations

Martin Hairer·Sam Punshon-Smith·Tommaso Rosati·Jaeyun Yi
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Abstract

We consider the top Lyapunov exponent associated to the advection-diffusion and linearised Navier-Stokes equations on the two-dimensional torus. The velocity field is given by the stochastic Navier-Stokes equations driven by a non-degenerate white-in-time noise with a power-law correlation structure. We show that the top Lyapunov exponent is bounded from below by a negative power of the diffusion parameter. This partially answers a conjecture of Doering and Miles and provides a first lower bound on the Batchelor scale in terms of the diffusivity. The proof relies on a robust analysis of the projective process associated to the linear equation, through its spectral median dynamics. We introduce a probabilistic argument to show that high-frequency states for the projective process are unstable under stochastic perturbations, leading to a Lyapunov drift condition and quantitative-in-diffusivity estimates.

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Martin Hairer, Sam Punshon-Smith, Tommaso Rosati, Jaeyun Yi. 2024-11-15. Lower bounds on the top Lyapunov exponent for linear PDEs driven by the 2D stochastic Navier-Stokes equations. https://arxiv.org/abs/2411.10419

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